called the Fibonacci
sequence and F is called the n-th Fibonacci numbers.

Fibonacci was one
of the greatest European mathematician of the middle ages, his full
name was Leonardo of Pisa, or Leonardo Pisano in Italian, he was born
in Pisa (Italy), the city with the famous Leaning Tower, about 1175
AD.

The Lucas numbersare defined by the equations ,(3)

and
satisfy the same recurrence
where, the first few are

1, 3, 4, 7, 11, 18, 29, 47, 76,123, ...

The
French mathematician, Edouard Lucas (1842-1891), was the first who
gave the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, .. the name the
Fibonacci Numbers.

Assuming that the
sequence Fn has the form
, where is a real parameter.

Substituting in (1)

(4)

or equivalent

But
("nN*), the last
equality becomes

This is a quadratic
equation with res pect to the real parameter l having the roots

and(5)

Thus the sequences
,

verify the equality
(1). So we conclude that the equation (1) can have more solutions. In
general there are an infinite of the sequences verifying (1). Easy to
observe that (1) has the form(6)

where c1,
c2 are fixed real numbers, verifying (1), too.
Also you can prove that any sequence verifying (1) has the form (6).

For n = 0 and
n = 1 in (6), we get the linear system

having the solutions

Finally, the general
term of the Fibonacci sequence has the form

N.(7)

Some Proprieties of
the Fibonacci sequence.

1.(8)

Proof:

.

Summing all
equalities we get
,

butso (8) isshown.

2.
.

3.
.

2. and 3.can be proven in a similar manner.

4.(9)

Proof: It is easy to
observe that
,
.

From this we have
successively the equalities:

.

Summing all these
equalities we get (9).

5.Prove that(10)

where Fn
is the n-th term of the Fibonacci sequences.

Proof: Using (1) and
(9) then (10) is easy to be proven.

But my goal is to
prove (10) using the mathematical induction. Ill proceed by
mathematical induction with respect
.

For m = 1,
the equality (10) becomes
, this being evident, then

(10) is true for
m=1. (For example when m = 2 the formula (10) is true

).

Assuming that (10)
is true for m = k and m = k + 1.

Ill prove that
(10) is true for m = k + 2, too.

Therefore, being
true the equalities

,

Summing both
equalities we get
,

in fact this is (10)
for m = k + 2.

6. Prove that(sometimes called double
angle formula),

Proof:If in(10)m = nwe get
,

7. Prove that
,n>1(11)

Proof: Again Ill
proceed for (11) by mathematical induction.

For n = 2
then (11) becomes
, which is true. Thus (11) is true for n=2.

Assuming that (11)
is true for n, Ill prove that it is true for n + 1,
too.